Graphs

Coordinate Geometry

Coordinate geometry is mathematics which unites geometry with algebra. Mathematician Rene Descartes (1596 – 1650) introduced the coordinate plane, a grid with numerical markings. Any point on the plane can be located by its values on the X and Y axes, called its coordinates. (This is like GPS coordinates in Google Maps.) Then geometric objects are laid on the plane, and algebraic functions are associated with those geometric objects. This innovation enabled us to do mathematics with either geometry or algebra as needs be.

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The simplest geometric object are graphs. A graph is a line drawn on the coordinate plane. The points on the line have the coordinates (x, y). The line is associated with this collection (or set) of (x,y) pair of values, called “coordinates”.

The ingenious concept is to choose the collection of points which OBEY an algebraic function y(x) and declare that the graph (line) also represent the function. So, to work with that function, you can choose to use algebra or geometry, and sometimes both.

It is important to note that ALL points on the graph (line) obeys the equation y(x). These are the ONLY points; no other points in the plane obeys the equation.

That means two requirements:

  1. If you take any point on the graph, its coordinates (x1, y1) would obey the function y(x). That means given x=x1, the function y(x1) would return y1.
  2. If you substitute the value x1 into function y(x) and it returns value y1 = y(x1); then (x1, y1) would be the coordinates of a point on the graph, and no where else.

There are two plots with one graph each in the picture above. The one on the left is a “straight line” graph, the equation is y(x) = x. The one on the right is a curved line, with equation y(x) = x^3 .

Task: Can you verify these functions? Can you show that the above 2 requirements are satisfied?

The graph and the formula are two representation of the relation between the variables y and x. Each individually is sufficient to describe the relation. However, if the relation changes, both representations have to change, in tandem. This is what we are considering next.

Note: It is important to realize that the xy graph is NOT a map. The x-axis is not pointing east and y-axis pointing north. The curve drawn on it is not a path on a map. The curve (graph) shows a relation between the x and y coordinates of points on it. It is not a “free path”, it is determined by the equation of the function y(x).

Changing the Graph coordinates

One way to test your understanding of graphs is to change relation between y and x. One common exercise is to show the change by shifting the graph, and determine how the function must change.

Before we do that, I will tell you a “secret”: there is another way to represent the relation between y and x; that is a table. Let us look at the table and the graph of the function y(x) = x^3 .x_3_all

The table shows a selection of values of x and y, and the graph draws the line (also called the locus). You can locate the tabled points on the graph.

You can check that the function for the y-x relation is y(x) = x^3 and that agrees with the table.

If the y-x relation change, its representations x_3+3_allwould all change.  These are some examples:

Consider the change of increasing the values of y by 3, without changing the values of x. The values in the data table will changed to the second picture.

Comparing the values in the data tables, it can be seen clearly that the values for Y would just be increased by 3, giving the equation y(x) = x^3 + 3.x+2_3_all

Consider another example of decreasing the values of x by 2, not changing the values of y. The data table would change to this third picture. Plotting the points in the new table would give the graph as shown. Comparing this with the first picture, we see that the graph (locus) has shifted to the left, towards the negative side of the X axis. A smaller (more negative) value of x gives back the same values of y as before. This is not surprising.

But what may be surprising is the change in the formula. The value x to send to the function y(x) is now smaller, but the function y(x) has to treat it AS IF it received the old x value in order for it to return the expected value of y. To do that the function does a “pre-processing” which transforms the new value to the old value. In this case, the pre-processing is ADDing 2 to the given x input. This change the formula for the changed relation:

It is now y(x) = (x + 2)^3

Memo: Try this easy problem: move line y(x) = x to the right by 3. Old equation has y = 0 when x = 0. New equation will have y = 0 when x = 3. Should it be “y = x+3” or “y = x-3”? It must be the latter. 🙂

Another way to to explain this makes use of “inverse functions“.